IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 6, JUNE 2014
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Multirate Repetitive Control for PWM DC/AC Converters Bin Zhang, Senior Member, IEEE, Keliang Zhou, Senior Member, IEEE, and Danwei Wang, Senior Member, IEEE
Abstract—In this paper, a multirate repetitive control (RC) scheme is developed and applied to constant-voltage constantfrequency pulsewidth modulation converter systems. In this scheme, the converter has a fast sampling rate, while the repetitive controller has a reduced sampling rate. The learning is based on the downsampled input and error signals in previous periods. The multirate RC synthesis method, as well as its convergence and stability conditions, is discussed in detail. Systematic experiments are also carried out to illustrate the effectiveness of multirate RC. Experimental results show that, with a well-designed multirate RC, the total harmonic distortion can be very low. This approach can reduce the computation delay caused by the plug-in RC in each switching control period and will enhance the system stability. Consequently, the switching control frequency of the converter can be increased to compensate the control performance loss. This approach is suitable for design of cost-effective and flexible converter control systems. Index Terms—Constant-voltage constant-frequency (CVCF) pulsewidth-modulation (PWM) converter, multirate repetitive control (RC), total harmonic distortion (THD).
I. I NTRODUCTION
C
ONSTANT-VOLTAGE constant-frequency (CVCF) pulsewidth-modulation (PWM) dc–ac converters are widely used in many ac power-conditioning systems. The periodic nature of dc–ac converter output voltage makes repetitive control (RC) [1] an effective way to achieve low total harmonic distortion (THD). RC has been successfully applied to uninterruptible power supply (UPS) [2], [3], converters [4]–[7], harmonic compensation [8]–[10], and power filters [11]–[13], among others. RC is based on the internal model principle, which states that a control system with a periodic signal generator of known period inside the closed loop can track any reference signal with the same period exactly. In many RC converter systems [14]–[19], the sampling frequency of a converter feedback control loop is chosen identical to the PWM switching frequency. There are some multirate RC schemes
Manuscript received January 9, 2013; revised April 16, 2013; accepted June 14, 2013. Date of publication July 23, 2013; date of current version December 20, 2013. B. Zhang is with the Department of Electrical Engineering, University of South Carolina, Columbia, SC 29208 USA (e-mail:
[email protected]). K. Zhou is with the Department of Electrical and Computer Engineering, University of Canterbury, Christchurch 8020, New Zealand (e-mail: eklzhou@ ieee.org). D. Wang is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2013.2274423
with downsampling/upsampling schemes in other areas [20], [21]. In this work, based on some previous works [4]–[7], a downsampled multirate scheme is developed and applied to CVCF PWM converters. In the proposed scheme, the converter has a fast sampling rate (termed as feedback rate), while RC has a slow sampling rate (termed as RC rate). Error and input signals of previous periods are downsampled to the RC rate before they are used in RC. The RC is implemented at the RC rate, and its output is upsampled to the feedback rate for converter control. The design of multirate RC, its convergence and stability conditions, and performance analysis are elaborated. By downsampling, the computation delay caused by the plug-in RC in each switching control period can be reduced. This will help to enhance the system stability and increase the switching control frequency of the converter. The performance loss caused by downsampling RC will then be compensated. The overall performance will not be significantly sacrificed. This enables us to design costeffective and flexible control schemes for converters. Systematical experiments are carried out to verify the effectiveness of the proposed scheme. The results show that multirate RC can simplify the online calculation, keep the convergence rate, and generate low THD with well-selected parameters. However, the peak error will become large under nonlinear load. II. M ULTIRATE RC A. Scheme of Multirate RC Fig. 1 shows the structure of the multirate RC system. In this figure, the converter feedback control system has a feedback rate with a sampling period of Tf = T , where Gs (z) is the plant, Gc (z) is the controller, D(z) is the disturbance, and E(z) = Yd (z) − Y (z) is the tracking error between the reference signal Yd (z) and plant output Y (z). It is assumed that the converter control system G(z) = (Gc (z)Gs (z)/1 + Gc (z)Gs (z)) is asymptotically stable. The plug-in repetitive controller Gr has an RC rate with a sampling period of Ts = mT . In the RC block, E(zm ) is the error signal passing through an antialiasing filter Fa,a and a downsampling process; kr is the RC gain; Gf (zm ) is often designed as the inverse of the closed-loop feedback system; Q(zm ) is a lowpass filter introduced to enhance robustness, which is often a −1 with first-order zero-phase filter Q(zm ) = a1 zm + a0 + a1 zm a0 + 2a1 = 1 [22]; and Ur (zm ) is the output of RC at the RC rate, which is interpolated by a zero-order holder H and filtered by an anti-imaging Fa,i to get its feedback rate counterpart Ur (z).
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 6, JUNE 2014
Fig. 1. Multirate RC system.
The ratio between the feedback rate and RC rate is m, which is termed as “sampling ratio” [20]. The relationship between the two rates can be expressed as Ts = mT ;
z = esT ;
zm = esTs = z m .
(1)
Note that a periodic signal generator of period Ns is inside Gr , which is given as P (zm ) =
−Ns zm 1 = Ns −Ns 1 − zm zm − 1
−Ns zm Q(zm ) Gf (zm ). −Ns 1 − zm Q(zm )
˜r is the signal after increasing the sampling In this equation, U ˜r by anti-imaging number, and Ur is the result of filtering U filter Fa,i . The RC output Ur (z) can then be denoted as
(2) Ur (z) = Fa,i Fzoh Gr Fdown Fa,a E(z).
where Ns = (fc /f )/m with f being the reference signal frequency and fc being the sampling frequency. According to the internal model principle, this RC system can achieve zero-error tracking of the periodic reference signal with period Ns . The transfer function of the repetitive controller Gr is Gr (zm ) = kr
anti-imaging filter Fa,i with a theoretical cutoff frequency of π/m to prevent the distortion of frequency spectra ˜r (mK + i) = Ur (K), i = 0, . . . , m − 1 Fzoh : U (5) ˜r (z). Ur (z) = Fa,i (z)U
(3)
When a signal is downsampled, the Nyquist criterion must still be satisfied with respect to the new sampling period to avoid aliasing. The RC with a sampling period of Ts downsamples the signal by m and causes aliasing of any frequencies in the original signal above |ω| > π/m. To prevent this aliasing, the signal is filtered by a low-pass antialiasing filter Fa,a , with appropriate cutoff frequency to reduce the bandwidth of the signal. Since a practical low-pass filter does not have perfect cutoff frequency, the cutoff frequency should be set below the theoretical cutoff frequency π/m. Then the downsampled error signal can be written as ˜ E(z) = Fa,a (z)E(z) (4) ˜ + 1) Fdown : E(K + 1) = E(mK ˜ is the error signal after antialiasing filtering. The where E second equation is a downsampling process with sampling ratio m with K being the sampling index. As mentioned earlier, the output of RC Ur (zm ) is at the RC rate. To use it for converter control, the signal must be upsampled to a signal Ur (z) at the feedback rate with a sampling period of T . To this end, a zero-order holder is used to hold the values between every two samples and then filtered by an
(6)
B. Equivalent Closed-Loop System To analyze the multirate RC system in Fig. 1, it is transformed to an equivalent system with a single sampling rate at ¯ s (zm ) ¯ c (zm ) and G the RC rate, as shown in Fig. 2, in which G are the RC rate counterparts of Gc (z) and Gs (z), respectively. To simplify the analysis, the antialiasing filter Fa,a and sampler are considered together. The effect of them can be absorbed by filter Gf to simplify the analysis. Similarly, the effect of antiimaging filter Fa,i and zero-order hold can be absorbed by Gf . For denotation simplicity, write closed-loop system G(z) in state-space form as follows: xf (k + 1) = Af xf (k) + Bf uf (k) (7) yf (k) = Cf xf (k) + vf (k) where xf , uf , yf , and vf are the state, input, output, and disturbance, respectively. For (7), states in an RC rate sampling period mT are ⎧ xf (Km+1) = Af xf (Km) + Bf uf (Km) ⎪ ⎪ ⎪ ⎪ xf (Km+2) = A2f xf (Km)+Af Bf uf (Km)+Bf uf (Km) ⎪ ⎨ .. . ⎪ ⎪ m ⎪ ⎪ (Km+m) = A x (Km)+Am−1 Bf uf (Km) + · · · x f f ⎪ f f ⎩ + Af Bf uf (Km) + Bf uf (Km). Then, by downsampling, its slow-rate state function is xs (K + 1)
m−1 us (K) = Am x (K) + A B + · · · + A B + B s f f f f f f = As xs (K) + Bs us (K)
m−1 where As = Am Bf + · · · + Af Bf + Bf ). f and Bs = (Af
ZHANG et al.: MULTIRATE REPETITIVE CONTROL FOR PWM DC/AC CONVERTERS
Fig. 2.
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Equivalent single-rate RC.
For the output, we have ys (K) = Cs xs (K) + vs (K) with Cs = Cf . The RC rate counterpart of (7) has the form of xs (K + 1) = As xs (K) + Bs us (K) (8) ys (K) = Cs xs (K) + vs (K). ¯ m) = The equivalent RC rate transfer function is G(z Cs (zm I − As )−1 Bs . In such a system, the overall transfer function from Yd (zm ) and D(zm ) to Y (zm ) can be respectively derived as follows:
−Ns ¯ m) 1 − Q(zm )zm (1 − kr Gf (zm )) G(z Y (zm ) = (9) −Ns ¯ m) Yd (zm ) 1 − kr Gf (zm )G(z 1 − Q(zm )zm −Ns ¯ m ) 1 − Q(zm )zm S(z Y (zm ) = (10) −Ns ¯ m) D(zm ) 1 − Q(zm )zm 1 − kr Gf (zm )G(z
C. Design of Gf (zm ) The filter Gf (zm ) is often designed as the inverse of the closed-loop feedback control system for zero-phase compensation. Due to the large load changes (particularly nonlinear loads) and various uncertainties, the inverse of the closedloop system, or even its good approximation, is usually not available. On the other hand, the linear phase compensation RC is simple and effective in offsetting phase lag and time delay. In this design, the multirate RC takes the form of linear phase compensation, and Gf (zm ) is designed as γ Gf (zm ) = zm
where γ is the lead step. Such a lead step γ will produce a linear phase lead θ=γ
¯ m ) is the RC rate counterpart of S(z) = (1/1 + where S(z Gc (z)Gs (z)). The error for the overall system is derived as follows: −Ns ¯ m) 1 − Q(zm )zm 1 − G(z E(zm ) = −Ns ¯ m) 1 − kr Gf (zm )G(z 1 − Q(zm )zm × (Yd (zm ) − D(zm )) .
(11)
From (9)–(11), the stability of the systems requires
¯ m ) < 1,
Q(zm ) 1 − kr Gf (zm )G(z
zm = ejωm ; π . ∀0 < ωm < Ts
(12)
If frequency ωm of reference yd (t) and disturbance d(t) approaches ωl = 2πlf with l = 0, 1, 2, . . . , Ls (Ls = Ns /2 for −Ns = 1. even Ns and Ls = (Ns − 1)/2 for odd Ns ), then zm From (11) with the assumption that Q(zm ) = 1, we have lim E(zm ) = 0;
ω→ωl
∀ωl .
(13)
Equation (13) indicates that this scheme can suppress all harmonics of the periodic reference signal up to Ls , even in the presence of unmodeled dynamics. Note that the single-rate RC is a special case of multirate RC when m = 1.
ω 180◦ ωN
at frequency ω, which reaches γ × 180◦ at Nyquist frequency ωN . With this Gf (zm ) filter, (12) is modified as
γ ¯
Q(zm ) 1 − kr zm G(zm ) < 1,
∀0 < ω
